The file ActionPotential.DAT contains two columns of data – time and voltage – showing a typical action potential in a sensory neuron. (The MATLAB command
load ActionPotential.DAT
will import the data into the workspace with the variable name ActionPotential.)
We are interested not only in the voltage, but also in total membrane current over the course of the action potential. We have not measured current directly, but can estimate it using the relationship
I =−C dV/dt
where C is membrane capacitance. C can be estimated from cell surface area using the approximation that 1 cm2 of membrane has a capacitance of 1µF.
Using a central difference approximation of order h2 to estimate the derivative of V, calculate the current I using the equation above when the cell diameter is 1 micron (that is, 1µm, or 10-6 m). Indicate the time points at which the central difference approximation cannot be applied, and use a forward or backward difference approximation (also of order h2), as appropriate, to calculate the derivative at those points. (Your result, therefore, should produce a value of the derivative at EVERY time point represented in the data set.)
Make two plots: one will be the voltage (y-axis) as a function of time (x-axis), and the other will be the estimated total membrane current (y-axis) as a function of voltage (xaxis). This first plot is a time-domain plot, and the second is a state-space plot. On both plots, label (for example, by adding symbols or text labels to the plot) the peaks (maximum and minimum voltage) of the action potential. On the state-space plot, also label the peak inward (negative) current and the peak outward (positive) current.
Because the membrane returns to the same voltage from which it started, the statespace plot forms a loop. If we were to draw arrows representing how we proceed around the loop with time, would those arrows proceed in a clockwise or a counterclockwise direction? How can you tell? What does this tell us about the current involved in creating an action potential – does the early part of the action potential start with mostly inward (negative) or mostly outward (positive) current?
(Action Potential data)
-1.92 -57.75
-1.905 -57.7
-1.89 -57.75
-1.875 -57.75
-1.86 -57.75
-1.845 -57.7
-1.83 -57.7
-1.815 -57.8
-1.8 -57.75
-1.785 -57.75
-1.77 -57.75
-1.755 -57.8
-1.74 -57.8
-1.725 -57.75
-1.71 -57.7
-1.695 -57.75
-1.68 -57.8
-1.665 -57.75
-1.65 -57.75
-1.635 -57.75
-1.62 -57.75
-1.605 -57.75
-1.59 -57.7
-1.575 -57.7
-1.56 -57.75
-1.545 -57.75
-1.53 -57.7
-1.515 -57.7
-1.5 -57.7
